A discontinuous Galerkin Reduced Basis Element method for elliptic problems

34/2014

A discontinuous Galerkin Reduced Basis Element method for elliptic problems

Friday 12th September 2014

Antonietti, P.F.; Pacciarini, P.; Quarteroni, A.

We propose and analyse a new discontinuous reduced basis element method for the approximation of parametrized elliptic PDEs in partitioned domains. The method is built upon an offline stage (parameter independent) and an online (parameter dependent) one. In the offline stage we build a non-conforming (discontinuous) global reduced space as a direct sum of local basis functions built independently on each subdomain. In the online stage, for a given value of the parameter, the approximate solution is obtained by ensuring the weak continuity of the fluxes and of the solution itself thanks to a discontinuous Galerkin approach. The new method extends and generalizes the methods introduced by L. Iapichino, G. Rozza and A. Quarteroni [Comput. Methods Appl. Mech. Engrg. 221/222 (2012), 63–82] and by L. Iapichino [PhD thesis, EPF Lausanne, 2012]. We prove stability and convergence properties of the proposed method, as well as conditioning properties of the associated algebraic online system. We also propose a two-level preconditioner for the online problem which exploits the pre-existing decomposition of the domain and is based upon the introduction of a global coarse finite element space. Numerical tests are performed to validate our theoretical results.

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ESAIM: Mathematical Modelling and Numerical Analysis